It is important to Leavers to refer to a "People's vote" as a second referendum as they wish to characterise it as an attempt to reverse the 2016 referendum which opens up several strong rhetorical cards, like "voting continually until to get the result you want." That could even be the most effective argument in a subsequent debate even though it is entirely irrelevant to the only important question which is determining the choice makes the country and it's people better off.

Similarly remainers should focus on another referendum being about making it clear which of various interpretations of "leaving' is the one you want, if any. I think referring to it as a third referendum strengthens the "vote until you get what you wanted" irrelevancy.